A084947 -id:A084947 - cp's OEIS frontend

A084942 Enneagorials: n-th polygorial for k=9.

1, 1, 9, 216, 9936, 745200, 82717200, 12738448800, 2598643555200, 678245967907200, 220429939569840000, 87290256069656640000, 41375581377017247360000, 23128949989752641274240000, 15056946443328969469530240000, 11292709832496727102147680000000, 9666559616617198399438414080000000

Offset: 0

Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003

Daniel Dockery, Polygorials, Special "Factorials" of Polygonal Numbers, preprint, 2003.

Cf. A006472, A001044, A000680, A084939, A084940, A084941, A084943, A084944, A085356, A220605.

a := n->n!/2^n*product(7*i+2,i=0..n-1); [seq(a(j),j=0..30)];

polygorial[k_, n_] := FullSimplify[ n!/2^n (k -2)^n*Pochhammer[2/(k -2), n]]; Array[polygorial[9, #] &, 16, 0] (* Robert G. Wilson v, Dec 26 2016 *)

a(n)=n!/2^n*prod(i=1,n,7*i-5) \\ Charles R Greathouse IV, Dec 13 2016

a(n) = polygorial(n, 9) = (A000142(n)/A000079(n))*A084947(n) = (n!/2^n)*Product_{i=0..n-1} (7*i+2) = (n!/2^n)*7^n*Pochhammer(2/7, n) = (n!/2^n)*7^n*Gamma(n+2/7)/Gamma(2/7).
D-finite with recurrence 2*a(n) = n*(7*n-5)*a(n-1). - R. J. Mathar, Mar 12 2019
a(n) ~ 7^n * n^(2*n + 2/7) * Pi /(Gamma(2/7) * 2^(n-1) * exp(2*n)). - Amiram Eldar, Aug 28 2025

A051188 Sept-factorial numbers.

Original entry on oeis.org

1, 7, 98, 2058, 57624, 2016840, 84707280, 4150656720, 232436776320, 14643516908160, 1025046183571200, 78928556134982400, 6629998715338521600, 603329883095805465600, 59126328543388935628800

Offset: 0

Wolfdieter Lang

For n >= 1, a(n) is the order of the wreath product of the symmetric group S_n and the Abelian group (C_7)^n. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 07 2001

Vincenzo Librandi, Table of n, a(n) for n = 0..300
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 614.
Index entries for sequences related to factorial numbers

Cf. A000142, A000165, A032031, A045754, A047053, A047058, A049209, A051186, A052562, A053106, A084947, A092516, A092750, A144827, A144739, A147585.

[7^n*Factorial(n): n in [0..20]]; // Vincenzo Librandi, Oct 05 2011

a(n) = n!*7^n; \\ Michel Marcus, Jun 08 2020

a(n) = n!*7^n =: (7*n)(!^7).
a(n) = 7*A034834(n) = Product_{k=1..n} 7*k, n >= 1.
E.g.f.: 1/(1 - 7*x).
G.f.: 1/(1 - 7*x/(1 - 7*x/(1 - 14*x/(1 - 14*x/(1 - 21*x/(1 - 21*x/(1 - 28*x/(1 - 28*x/(1 - ... (continued fraction). - Philippe Deléham, Jan 08 2012
From Amiram Eldar, Jun 25 2020: (Start)
Sum_{n>=0} 1/a(n) = e^(1/7) (A092516).
Sum_{n>=0} (-1)^n/a(n) = e^(-1/7) (A092750). (End)

A114799 Septuple factorial, 7-factorial, n!7, n!!!!!!!, a(n) = n*a(n-7) if n > 1, else 1.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 6, 7, 8, 18, 30, 44, 60, 78, 98, 120, 288, 510, 792, 1140, 1560, 2058, 2640, 6624, 12240, 19800, 29640, 42120, 57624, 76560, 198720, 379440, 633600, 978120, 1432080, 2016840, 2756160, 7352640, 14418720, 24710400, 39124800

Offset: 0

Jonathan Vos Post, Feb 18 2006

Many of the terms yield multifactorial primes a(n) + 1, e.g.: a(2) + 1 = 3, a(4) + 1 = 5, a(6) + 1 = 7, a(9) + 1 = 19, a(10) + 1 = 31, a(12) + 1 = 61, a(13) + 1 = 79, a(24) + 1 = 12241, a(25) + 1 = 19801, a(26) + 1 = 29641, a(29) + 1 = 76561, a(31) + 1 = 379441, a(35) + 1 = 2016841, a(36) + 1 = 2756161, ...

Equivalently, product of all positive integers <= n congruent to n (mod 7). - M. F. Hasler, Feb 23 2018

			a(40) = 40 * a(40-7) = 40 * a(33) = 40 * (33*a(26)) = 40 * 33 * (26*a(19)) = 40 * 33 * 26 * (19*a(12)) = 40 * 33 * 26 * 19 * (12*a(5)) = 40 * 33 * 26 * 19 * 12 5 = 39124800.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Multifactorial.
Index entries for sequences related to factorial numbers

Cf. A045754, A084947, A288094.

Cf. k-fold factorials: A000142, A001147 (and A000165, A006882), A007559 (and A032031, A008544, A007661), A007696 (and A001813, A008545, A047053, A007662), A008548 (and A052562, A047055, A085157), A085158 (and A008542, A047058, A047657), A045755.

a:= function(n)
    if n<1 then return 1;
    else return n*a(n-7);
    fi;
  end;
List([0..40], n-> a(n) ); # G. C. Greubel, Aug 20 2019

b:= func< n | (n lt 8) select n else n*Self(n-7) >;
[1] cat [b(n): n in [1..40]]; // G. C. Greubel, Aug 20 2019

A114799 := proc(n)
    option remember;
    if n < 1 then
        1;
    else
        n*procname(n-7) ;
    end if;
end proc:
seq(A114799(n),n=0..40) ; # R. J. Mathar, Jun 23 2014
A114799 := n -> product(n-7*k,k=0..(n-1)/7); # M. F. Hasler, Feb 23 2018

a[n_]:= If[n<1, 1, n*a[n-7]]; Table[a[n], {n,0,40}] (* G. C. Greubel, Aug 20 2019 *)

A114799(n,k=7)=prod(j=0,(n-1)\k,n-j*k) \\ M. F. Hasler, Feb 23 2018

def a(n):
    if (n<1): return 1
    else: return n*a(n-7)
[a(n) for n in (0..40)] # G. C. Greubel, Aug 20 2019

a(n) = 1 for n <= 1, else a(n) = n*a(n-7).

Sum_{n>=0} 1/a(n) = A288094. - Amiram Eldar, Nov 10 2020

Edited by M. F. Hasler, Feb 23 2018

A144739 7-factorial numbers A114799(7n+3): Partial products of A017017(k) = 7k+3, a(0) = 1.

Original entry on oeis.org

1, 3, 30, 510, 12240, 379440, 14418720, 648842400, 33739804800, 1990648483200, 131382799891200, 9590944392057600, 767275551364608000, 66752972968720896000, 6274779459059764224000, 633752725365036186624000, 68445294339423908155392000, 7871208849033749437870080000

Offset: 0

Philippe Deléham, Sep 20 2008

nonn

			a(0)=1, a(1)=3, a(2)=3*10=30, a(3)=3*10*17=510, a(4)=3*10*17*24=12240, ...

G. C. Greubel, Table of n, a(n) for n = 0..335
Index entries for sequences related to factorial numbers.

Cf. A114799, A001710, A001147, A032031, A008545, A047056, A011781, A045754, A084947, A144827, A147585, A049209, A051188.

List([0..20], n-> Product([0..n-1], k-> 7*k+3) ); # G. C. Greubel, Aug 19 2019

[ 1 ] cat [ &*[ (7*k+3): k in [0..n] ]: n in [0..20] ]; // Klaus Brockhaus, Nov 10 2008

a:= n-> product(7*j+3, j=0..n-1); seq(a(n), n=0..20); # G. C. Greubel, Aug 19 2019

Table[7^n*Pochhammer[3/7, n], {n,0,20}] (* G. C. Greubel, Aug 19 2019 *)

a(n)=prod(i=1,n,7*i-4) \\ Charles R Greathouse IV, Jul 02 2013

[product(7*k+3 for k in (0..n-1)) for n in (0..20)] # G. C. Greubel, Aug 19 2019

a(n) = Sum_{k=0..n} A132393(n,k)*3^k*7^(n-k).
G.f.: 1/(1-3*x/(1-7*x/(1-10*x/(1-14*x/(1-17*x/(1-21*x/(1-24*x/(1-... (continued fraction). - Philippe Deléham, Jan 08 2012
a(n) = (-4)^n*Sum_{k=0..n} (7/4)^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
From Ilya Gutkovskiy, Mar 23 2017: (Start)
E.g.f.: 1/(1 - 7*x)^(3/7).
a(n) ~ sqrt(2*Pi)*7^n*n^n/(exp(n)*n^(1/14)*Gamma(3/7)). (End)
a(n) = A114799(7*n-4). - M. F. Hasler, Feb 23 2018
D-finite with recurrence: a(n) +(-7*n+4)*a(n-1)=0. - R. J. Mathar, Feb 21 2020
Sum_{n>=0} 1/a(n) = 1 + (e/7^4)^(1/7)*(Gamma(3/7) - Gamma(3/7, 1/7)). - Amiram Eldar, Dec 19 2022

A084948 a(n) = Product_{i=0..n-1} (8*i+2).

Original entry on oeis.org

1, 2, 20, 360, 9360, 318240, 13366080, 668304000, 38761632000, 2558267712000, 189311810688000, 15523568476416000, 1397121162877440000, 136917873961989120000, 14513294639970846720000, 1654515588956676526080000, 201850901852714536181760000, 26240617240852889703628800000

Offset: 0

Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003

G. C. Greubel, Table of n, a(n) for n = 0..330

Cf. A000079, A000142, A000165, A008544, A001813, A047055, A047657, A048994, A084943, A084947, A084949.

List([0..20], n-> Product([0..n-1], k-> 8*k+2) ); # G. C. Greubel, Aug 18 2019

[1] cat [(&*[8*k+2: k in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Aug 18 2019

a := n->product(8*i+2,i=0..n-1); [seq(a(j),j=0..30)];

Table[8^n*Pochhammer[1/4, n], {n,0,20}] (* G. C. Greubel, Aug 18 2019 *)

vector(20, n, n--; prod(k=0, n-1, 8*k+2)) \\ G. C. Greubel, Aug 18 2019

[product(8*k+2 for k in (0..n-1)) for n in (0..20)] # G. C. Greubel, Aug 18 2019

a(n) = A084943(n)/A000142(n)*A000079(n) = 8^n*Pochhammer(1/4, n) = 1/2*Gamma(n+1/4)*sqrt(2)*Gamma(3/4)*8^n/Pi.
a(n) = (-6)^n*Sum_{k=0..n} (4/3)^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
G.f.: 2/G(0), where G(k)= 1 + 1/(1 - 2*x*(8*k+2)/(2*x*(8*k+2) - 1 + 16*x*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 30 2013
From Ilya Gutkovskiy, Mar 23 2017: (Start)
E.g.f.: 1/(1 - 8*x)^(1/4).
a(n) ~ sqrt(2*Pi)*8^n*n^n/(exp(n)*n^(1/4)*Gamma(1/4)). (End)
Sum_{n>=0} 1/a(n) = 1 + (e/8^6)^(1/8)*(Gamma(1/4) - Gamma(1/4, 1/8)). - Amiram Eldar, Dec 20 2022

A084949 a(n) = Product_{i=0..n-1} (9*i+2).

Original entry on oeis.org

1, 2, 22, 440, 12760, 484880, 22789360, 1276204160, 82953270400, 6138542009600, 509498986796800, 46873906785305600, 4734264585315865600, 520769104384745216000, 61971523421784680704000, 7932354997988439130112000, 1086732634724416160825344000, 158662964669764759480500224000

Offset: 0

Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003

Robert Israel, Table of n, a(n) for n = 0..329

Cf. A000165, A008544, A001813, A047055, A047657, A084947, A084948.

Cf. A000079, A000142, A035012, A048994, A084944.

List([0..20], n-> Product([0..n-1], k-> 9*k+2) ); # G. C. Greubel, Aug 19 2019

[1] cat [(&*[9*k+2: k in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Aug 19 2019

a:= n-> product(9*i+2,i=0..n-1); seq(a(j),j=0..20);

Table[9^n*Pochhammer[2/9, n], {n,0,20}] (* G. C. Greubel, Aug 19 2019 *)

vector(20, n, n--; prod(k=0, n-1, 9*k+2)) \\ G. C. Greubel, Aug 19 2019

[product(9*k+2 for k in (0..n-1)) for n in (0..20)] # G. C. Greubel, Aug 19 2019

a(n) = A084944(n)/A000142(n)*A000079(n) = 9^n*Pochhammer(2/9, n) = 9^n*Gamma(n+2/9)/Gamma(2/9).
a(n) = (-7)^n*Sum_{k=0..n} (9/7)^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
E.g.f.: (1-9*x)^(-2/9). - Robert Israel, Mar 22 2017
D-finite with recurrence: a(n) + (-9*n+7)*a(n-1) = 0. - R. J. Mathar, Jan 20 2020
Sum_{n>=0} 1/a(n) = 1 + (e/9^7)^(1/9)*(Gamma(2/9) - Gamma(2/9, 1/9)). - Amiram Eldar, Dec 21 2022
a(n) ~ sqrt(2*Pi) * (9/e)^n * n^(n-5/18) / Gamma(2/9). - Amiram Eldar, Aug 30 2025

A144827 Partial products of successive terms of A017029; a(0)=1.

Original entry on oeis.org

1, 4, 44, 792, 19800, 633600, 24710400, 1136678400, 60243955200, 3614637312000, 242180699904000, 17921371792896000, 1451631115224576000, 127743538139762688000, 12135636123277455360000, 1237834884574300446720000, 134924002418598748692480000, 15651184280557454848327680000

Offset: 0

Philippe Deléham, Sep 21 2008

nonn

			a(0)=1, a(1)=4, a(2)=4*11=44, a(3)=4*11*18=792, a(4)=4*11*18*25=19800, ...

T. D. Noe, Table of n, a(n) for n = 0..100

Cf. A001715, A002866, A007559, A008546, A045754, A047053, A132393.

Cf. A049209, A049308, A051188, A084947, A144739, A147585.

[ 1 ] cat [ &*[ (7*k+4): k in [0..n] ]: n in [0..14] ]; // Klaus Brockhaus, Nov 10 2008

FoldList[Times,1,Range[4,150,7]] (* Harvey P. Dale, Apr 25 2014 *)

[1]+[4*7^(n-1)*rising_factorial(11/7, n-1) for n in (1..30)] # G. C. Greubel, Feb 22 2022

a(n) = Sum_{k=0..n} A132393(n,k)*4^k*7^(n-k).
G.f.: 1/(1-4*x/(1-7*x/(1-11*x/(1-14*x/(1-18*x/(1-21*x/(1-25*x/(1-... (continued fraction). - Philippe Deléham, Jan 08 2012
a(n) = (-3)^n*Sum_{k=0..n} (7/3)^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
From Ilya Gutkovskiy, Mar 23 2017: (Start)
E.g.f.: 1/(1 - 7*x)^(4/7).
a(n) ~ sqrt(2*Pi)*7^n*n^(n+1/14)/(exp(n)*Gamma(4/7)). (End)
a(n) = 4*7^(n-1)*Pochhammer(n-1, 11/7) with a(0) = 1. - G. C. Greubel, Feb 22 2022
Sum_{n>=0} 1/a(n) = 1 + (e/7^3)^(1/7)*(Gamma(4/7) - Gamma(4/7, 1/7)). - Amiram Eldar, Dec 19 2022

Corrected a(9) by Vincenzo Librandi, Jul 14 2011

A034829 a(n) = n-th sept-factorial number divided by 2.

Original entry on oeis.org

1, 9, 144, 3312, 99360, 3676320, 161758080, 8249662080, 478480400640, 31101226041600, 2239288274995200, 176903773724620800, 15213724540317388800, 1414876382249517158400, 141487638224951715840000, 15139177290069833594880000, 1725866211067961029816320000

Offset: 1

Wolfdieter Lang

Peter Luschny, Multifactorials.
Index entries for sequences related to factorial numbers.

Cf. A045754, A034830, A034831, A034832, A034833, A034834, A084947.

Drop[With[{nn = 50}, CoefficientList[Series[(-1 + (1 - 7*x)^(-2/7))/2, {x, 0, nn}], x]*Range[0, nn]!], 1] (* G. C. Greubel, Feb 23 2018 *)

vector(20, n, prod(j=1, n, 7*j-5)/2) \\ Michel Marcus, Jan 07 2015

2*a(n) = (7*n-5)(!^7) = Product_{j=1..n} (7*j-5).
E.g.f.: (-1 + (1-7*x)^(-2/7))/2.
D-finite with recurrence: a(n) +(-7*n+5)*a(n-1)=0. - R. J. Mathar, Feb 24 2020
From Amiram Eldar, Dec 19 2022: (Start)
a(n) = A084947(n)/2.
Sum_{n>=1} 1/a(n) = 2*(e/7^5)^(1/7)*(Gamma(2/7) - Gamma(2/7, 1/7)). (End)

A147585 a(1) = 1; a(n) = (7n-9)a(n-1) for n > 1.

Original entry on oeis.org

1, 5, 60, 1140, 29640, 978120, 39124800, 1838865600, 99298742400, 6057223286400, 411891183475200, 30891838760640000, 2533130778372480000, 225448639275150720000, 21643069370414469120000, 2229236145152690319360000, 245215975966795935129600000, 28690269188115124410163200000

Offset: 1

Vladimir Joseph Stephan Orlovsky, Nov 08 2008

nonn

G. C. Greubel, Table of n, a(n) for n = 1..340

Cf. A048994, A132393.

Cf. A045754, A049209, A084947, A144739, A144827, A051188.

[ n eq 1 select 1 else Self(n-1)*(7*n-9): n in [1..15] ]; // Klaus Brockhaus, Nov 10 2008

[ 1 ] cat [ &*[ (5+7*k): k in [0..n-1] ]: n in [1..14] ]; // Klaus Brockhaus, Nov 10 2008

seq( -7^n*pochhammer(-2/7, n)/2, n = 1..15); # G. C. Greubel, Dec 03 2019

Table[-7^n*Pochhammer[-2/7, n]/2, {n, 15}] (* G. C. Greubel, Dec 03 2019 *)

{for(n=1, 15, print1(prod(k=1, n-1, 7*k-2,), ","))} \\ Klaus Brockhaus, Nov 10 2008

[-7^n*rising_factorial(-2/7, n)/2 for n in (1..15)] # G. C. Greubel, Dec 03 2019

a(n) = Product_{k=1..n-1} (7*k - 2). - Klaus Brockhaus, Nov 10 2008
a(n) = (5*7^(n-1)*Gamma(5/7+n))/Gamma(12/7). - Klaus Brockhaus, Nov 10 2008
a(n+1) = Sum_{k=0..n} A132393(n,k)*5^k*7^(n-k). - Philippe Deléham, Nov 09 2008
G.f.: x/(1-5x/(1-7x/(1-12x/(1-14x/(1-19x/(1-21x/(1-26x/(1-... (continued fraction). - Philippe Deléham, Jan 08 2012
a(n) = (-2)^n*Sum_{k=0..n} (7/2)^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
Sum_{n>=1} 1/a(n) = 1 + (e/7^2)^(1/7)*(Gamma(5/7) - Gamma(5/7, 1/7)). - Amiram Eldar, Dec 19 2022

Edited by Klaus Brockhaus, Nov 10 2008

A034832 a(n) = n-th sept-factorial number divided by 5.

Original entry on oeis.org

1, 12, 228, 5928, 195624, 7824960, 367773120, 19859748480, 1211444657280, 82378236695040, 6178367752128000, 506626155674496000, 45089727855030144000, 4328613874082893824000, 445847229030538063872000, 49043195193359187025920000, 5738053837623024882032640000

Offset: 1

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 1..339
Index entries for sequences related to factorial numbers.

Cf. A045754, A034829, A034830, A034831, A034833, A034834, A084947, A147585.

Rest[FoldList[Times,1,7*Range[20]-2]/5] (* Harvey P. Dale, May 30 2013 *)
Drop[With[{nn = 50}, CoefficientList[Series[(-1 + (1 - 7*x)^(-5/7))/5, {x, 0, nn}], x]*Range[0, nn]!], 1] (* G. C. Greubel, Feb 22 2018 *)

my(x='x+O('x^30)); Vec(serlaplace((-1 + (1-7*x)^(-5/7))/5)) \\ G. C. Greubel, Feb 22 2018

5*a(n) = (7*n-2)(!^7) = Product_{j=1..n} (7*j-2).
E.g.f.: (-1 + (1-7*x)^(-5/7))/5.
From Amiram Eldar, Dec 20 2022: (Start)
a(n) = A147585(n+1)/5.
Sum_{n>=1} 1/a(n) = 5*(e/7^2)^(1/7)*(Gamma(5/7) - Gamma(5/7, 1/7)). (End)

More terms from G. C. Greubel, Feb 22 2018

cp's OEIS Frontend

A084942 Enneagorials: n-th polygorial for k=9.

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A051188 Sept-factorial numbers.

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A114799 Septuple factorial, 7-factorial, n!7, n!!!!!!!, a(n) = n*a(n-7) if n > 1, else 1.

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A144739 7-factorial numbers A114799(7*n+3): Partial products of A017017(k) = 7*k+3, a(0) = 1.

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A084948 a(n) = Product_{i=0..n-1} (8*i+2).

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A084949 a(n) = Product_{i=0..n-1} (9*i+2).

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A144739 7-factorial numbers A114799(7n+3): Partial products of A017017(k) = 7k+3, a(0) = 1.

A147585 a(1) = 1; a(n) = (7n-9)a(n-1) for n > 1.